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Mirrors > Home > ILE Home > Th. List > elfz2 | Unicode version |
Description: Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show and . (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
elfz2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anass 399 | . 2 | |
2 | df-3an 965 | . . 3 | |
3 | 2 | anbi1i 454 | . 2 |
4 | df-fz 9920 | . . . 4 | |
5 | 4 | elmpocl 6021 | . . 3 |
6 | simpl 108 | . . 3 | |
7 | elfz1 9924 | . . . 4 | |
8 | 3anass 967 | . . . . 5 | |
9 | ibar 299 | . . . . 5 | |
10 | 8, 9 | syl5bb 191 | . . . 4 |
11 | 7, 10 | bitrd 187 | . . 3 |
12 | 5, 6, 11 | pm5.21nii 694 | . 2 |
13 | 1, 3, 12 | 3bitr4ri 212 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 w3a 963 wcel 2128 crab 2439 class class class wbr 3967 (class class class)co 5827 cle 7916 cz 9173 cfz 9919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-setind 4499 ax-cnex 7826 ax-resscn 7827 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4029 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-iota 5138 df-fun 5175 df-fv 5181 df-ov 5830 df-oprab 5831 df-mpo 5832 df-neg 8054 df-z 9174 df-fz 9920 |
This theorem is referenced by: elfz4 9928 elfzuzb 9929 uzsubsubfz 9956 fzmmmeqm 9967 fzpreddisj 9980 elfz1b 9999 fzp1nel 10013 elfz0ubfz0 10034 elfz0fzfz0 10035 fz0fzelfz0 10036 fz0fzdiffz0 10039 elfzmlbp 10041 fzind2 10148 iseqf1olemqcl 10395 iseqf1olemnab 10397 iseqf1olemab 10398 seq3f1olemqsumkj 10407 seq3f1olemqsumk 10408 summodclem2a 11290 fsum3 11296 fsum3cvg3 11305 fsumcl2lem 11307 fsumadd 11315 fsummulc2 11357 prodmodclem3 11484 prodmodclem2a 11485 fprodntrivap 11493 fprodeq0 11526 |
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